LMFDB - Embedded newform 1792.2.e.i.895.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(895,1792)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.7
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.i.895.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28825i q^{3} -0.540182 q^{5} +(1.41421 + 2.23607i) q^{7} -2.23607 q^{9} +O(q^{10})$	$q+2.28825i q^{3} -0.540182 q^{5} +(1.41421 + 2.23607i) q^{7} -2.23607 q^{9} -1.23607 q^{11} +2.28825 q^{13} -1.23607i q^{15} -1.74806i q^{17} +5.11667i q^{19} +(-5.11667 + 3.23607i) q^{21} +3.23607i q^{23} -4.70820 q^{25} +1.74806i q^{27} +2.00000i q^{29} +3.90879 q^{31} -2.82843i q^{33} +(-0.763932 - 1.20788i) q^{35} +6.94427i q^{37} +5.23607i q^{39} -10.2333i q^{41} -11.7082 q^{43} +1.20788 q^{45} +10.9010 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.00000 q^{51} +8.47214i q^{53} +0.667701 q^{55} -11.7082 q^{57} +1.62054i q^{59} -13.1893 q^{61} +(-3.16228 - 5.00000i) q^{63} -1.23607 q^{65} +3.70820 q^{67} -7.40492 q^{69} -10.0000i q^{71} -14.1421i q^{73} -10.7735i q^{75} +(-1.74806 - 2.76393i) q^{77} +3.52786i q^{79} -10.7082 q^{81} -9.02546i q^{83} +0.944272i q^{85} -4.57649 q^{87} +15.4775i q^{89} +(3.23607 + 5.11667i) q^{91} +8.94427i q^{93} -2.76393i q^{95} +1.74806i q^{97} +2.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} + 16 q^{25} - 24 q^{35} - 40 q^{43} - 24 q^{49} + 32 q^{51} - 40 q^{57} + 8 q^{65} - 24 q^{67} - 32 q^{81} + 8 q^{91} + 40 q^{99}+O(q^{100}) $ 8 * q + 8 * q^11 + 16 * q^25 - 24 * q^35 - 40 * q^43 - 24 * q^49 + 32 * q^51 - 40 * q^57 + 8 * q^65 - 24 * q^67 - 32 * q^81 + 8 * q^91 + 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			2.28825i			1.32112i	0.750774	+	0.660560i	$0.229681\pi$
$3$			2.28825i			1.32112i	−0.750774	+	0.660560i	$0.770319\pi$
$4$		0			0
$5$	−0.540182			−0.241577			−0.120788	−	0.992678i	$-0.538542\pi$
$5$	−0.540182			−0.241577			−0.120788	+	0.992678i	$0.538542\pi$
$6$		0			0
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$8$		0			0
$9$	−2.23607			−0.745356
$10$		0			0
$11$	−1.23607			−0.372689			−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607			−0.372689			−0.186344	+	0.982485i	$0.559664\pi$
$12$		0			0
$13$	2.28825			0.634645			0.317323	−	0.948318i	$-0.397216\pi$
$13$	2.28825			0.634645			0.317323	+	0.948318i	$0.397216\pi$
$14$		0			0
$15$		−	1.23607i		−	0.319151i
$16$		0			0
$17$		−	1.74806i		−	0.423968i	−0.977273	−	0.211984i	$-0.932008\pi$
$17$		−	1.74806i		−	0.423968i	0.977273	−	0.211984i	$-0.0679924\pi$
$18$		0			0
$19$			5.11667i			1.17385i	0.809643	+	0.586923i	$0.199661\pi$
$19$			5.11667i			1.17385i	−0.809643	+	0.586923i	$0.800339\pi$
$20$		0			0
$21$	−5.11667	+	3.23607i	−1.11655	+	0.706168i
$22$		0			0
$23$			3.23607i			0.674767i	0.941367	+	0.337383i	$0.109542\pi$
$23$			3.23607i			0.674767i	−0.941367	+	0.337383i	$0.890458\pi$
$24$		0			0
$25$	−4.70820			−0.941641
$26$		0			0
$27$			1.74806i			0.336415i
$28$		0			0
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$	3.90879			0.702039			0.351020	−	0.936368i	$-0.385835\pi$
$31$	3.90879			0.702039			0.351020	+	0.936368i	$0.385835\pi$
$32$		0			0
$33$		−	2.82843i		−	0.492366i
$34$		0			0
$35$	−0.763932	−	1.20788i	−0.129128	−	0.204169i
$36$		0			0
$37$			6.94427i			1.14163i	0.821078	+	0.570816i	$0.193373\pi$
$37$			6.94427i			1.14163i	−0.821078	+	0.570816i	$0.806627\pi$
$38$		0			0
$39$			5.23607i			0.838442i
$40$		0			0
$41$		−	10.2333i		−	1.59818i	−0.601211	−	0.799090i	$-0.705315\pi$
$41$		−	10.2333i		−	1.59818i	0.601211	−	0.799090i	$-0.294685\pi$
$42$		0			0
$43$	−11.7082			−1.78548			−0.892742	−	0.450568i	$-0.851222\pi$
$43$	−11.7082			−1.78548			−0.892742	+	0.450568i	$0.851222\pi$
$44$		0			0
$45$	1.20788			0.180061
$46$		0			0
$47$	10.9010			1.59008			0.795041	−	0.606556i	$-0.207450\pi$
$47$	10.9010			1.59008			0.795041	+	0.606556i	$0.207450\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$			8.47214i			1.16374i	0.813283	+	0.581869i	$0.197678\pi$
$53$			8.47214i			1.16374i	−0.813283	+	0.581869i	$0.802322\pi$
$54$		0			0
$55$	0.667701			0.0900328
$56$		0			0
$57$	−11.7082			−1.55079
$58$		0			0
$59$			1.62054i			0.210977i	0.994421	+	0.105488i	$0.0336406\pi$
$59$			1.62054i			0.210977i	−0.994421	+	0.105488i	$0.966359\pi$
$60$		0			0
$61$	−13.1893			−1.68872			−0.844358	−	0.535780i	$-0.820018\pi$
$61$	−13.1893			−1.68872			−0.844358	+	0.535780i	$0.820018\pi$
$62$		0			0
$63$	−3.16228	−	5.00000i	−0.398410	−	0.629941i
$64$		0			0
$65$	−1.23607			−0.153315
$66$		0			0
$67$	3.70820			0.453029			0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820			0.453029			0.226515	+	0.974008i	$0.427267\pi$
$68$		0			0
$69$	−7.40492			−0.891447
$70$		0			0
$71$		−	10.0000i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$		−	10.0000i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$		0			0
$73$		−	14.1421i		−	1.65521i	−0.561310	−	0.827606i	$-0.689702\pi$
$73$		−	14.1421i		−	1.65521i	0.561310	−	0.827606i	$-0.310298\pi$
$74$		0			0
$75$		−	10.7735i		−	1.24402i
$76$		0			0
$77$	−1.74806	−	2.76393i	−0.199210	−	0.314979i
$78$		0			0
$79$			3.52786i			0.396916i	0.980109	+	0.198458i	$0.0635933\pi$
$79$			3.52786i			0.396916i	−0.980109	+	0.198458i	$0.936407\pi$
$80$		0			0
$81$	−10.7082			−1.18980
$82$		0			0
$83$		−	9.02546i		−	0.990673i	−0.868701	−	0.495337i	$-0.835045\pi$
$83$		−	9.02546i		−	0.990673i	0.868701	−	0.495337i	$-0.164955\pi$
$84$		0			0
$85$			0.944272i			0.102421i
$86$		0			0
$87$	−4.57649			−0.490651
$88$		0			0
$89$			15.4775i			1.64062i	0.571922	+	0.820308i	$0.306198\pi$
$89$			15.4775i			1.64062i	−0.571922	+	0.820308i	$0.693802\pi$
$90$		0			0
$91$	3.23607	+	5.11667i	0.339232	+	0.536373i
$92$		0			0
$93$			8.94427i			0.927478i
$94$		0			0
$95$		−	2.76393i		−	0.283573i
$96$		0			0
$97$			1.74806i			0.177489i	0.996054	+	0.0887445i	$0.0282855\pi$
$97$			1.74806i			0.177489i	−0.996054	+	0.0887445i	$0.971715\pi$
$98$		0			0
$99$	2.76393			0.277786
$100$		0			0
$101$	11.4412			1.13844			0.569222	−	0.822184i	$-0.307244\pi$
$101$	11.4412			1.13844			0.569222	+	0.822184i	$0.307244\pi$
$102$		0			0
$103$	−13.7295			−1.35281			−0.676403	−	0.736532i	$-0.736462\pi$
$103$	−13.7295			−1.35281			−0.676403	+	0.736532i	$0.736462\pi$
$104$		0			0
$105$	2.76393	−	1.74806i	0.269732	−	0.170594i
$106$		0			0
$107$	0.291796			0.0282090			0.0141045	−	0.999901i	$-0.495510\pi$
$107$	0.291796			0.0282090			0.0141045	+	0.999901i	$0.495510\pi$
$108$		0			0
$109$		−	3.52786i		−	0.337908i	−0.985624	−	0.168954i	$-0.945961\pi$
$109$		−	3.52786i		−	0.337908i	0.985624	−	0.168954i	$-0.0540389\pi$
$110$		0			0
$111$	−15.8902			−1.50823
$112$		0			0
$113$	2.29180			0.215594			0.107797	−	0.994173i	$-0.465620\pi$
$113$	2.29180			0.215594			0.107797	+	0.994173i	$0.465620\pi$
$114$		0			0
$115$		−	1.74806i		−	0.163008i
$116$		0			0
$117$	−5.11667			−0.473037
$118$		0			0
$119$	3.90879	−	2.47214i	0.358318	−	0.226620i
$120$		0			0
$121$	−9.47214			−0.861103
$122$		0			0
$123$	23.4164			2.11139
$124$		0			0
$125$	5.24419			0.469055
$126$		0			0
$127$			0.763932i			0.0677880i	0.999425	+	0.0338940i	$0.0107909\pi$
$127$			0.763932i			0.0677880i	−0.999425	+	0.0338940i	$0.989209\pi$
$128$		0			0
$129$		−	26.7912i		−	2.35884i
$130$		0			0
$131$			5.78437i			0.505383i	0.967547	+	0.252692i	$0.0813158\pi$
$131$			5.78437i			0.505383i	−0.967547	+	0.252692i	$0.918684\pi$
$132$		0			0
$133$	−11.4412	+	7.23607i	−0.992080	+	0.627447i
$134$		0			0
$135$		−	0.944272i		−	0.0812700i
$136$		0			0
$137$	12.4721			1.06557			0.532783	−	0.846252i	$-0.321146\pi$
$137$	12.4721			1.06557			0.532783	+	0.846252i	$0.321146\pi$
$138$		0			0
$139$		−	2.70091i		−	0.229088i	−0.993418	−	0.114544i	$-0.963459\pi$
$139$		−	2.70091i		−	0.229088i	0.993418	−	0.114544i	$-0.0365407\pi$
$140$		0			0
$141$			24.9443i			2.10069i
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$		−	1.08036i		−	0.0897193i
$146$		0			0
$147$	−14.4721	−	6.86474i	−1.19364	−	0.566194i
$148$		0			0
$149$		−	13.4164i		−	1.09911i	−0.835456	−	0.549557i	$-0.814796\pi$
$149$		−	13.4164i		−	1.09911i	0.835456	−	0.549557i	$-0.185204\pi$
$150$		0			0
$151$			11.2361i			0.914378i	0.889369	+	0.457189i	$0.151144\pi$
$151$			11.2361i			0.914378i	−0.889369	+	0.457189i	$0.848856\pi$
$152$		0			0
$153$			3.90879i			0.316007i
$154$		0			0
$155$	−2.11146			−0.169596
$156$		0			0
$157$	11.4412			0.913109			0.456555	−	0.889695i	$-0.349083\pi$
$157$	11.4412			0.913109			0.456555	+	0.889695i	$0.349083\pi$
$158$		0			0
$159$	−19.3863			−1.53744
$160$		0			0
$161$	−7.23607	+	4.57649i	−0.570282	+	0.360678i
$162$		0			0
$163$	17.2361			1.35003			0.675017	−	0.737803i	$-0.264136\pi$
$163$	17.2361			1.35003			0.675017	+	0.737803i	$0.264136\pi$
$164$		0			0
$165$			1.52786i			0.118944i
$166$		0			0
$167$	8.07262			0.624678			0.312339	−	0.949971i	$-0.398888\pi$
$167$	8.07262			0.624678			0.312339	+	0.949971i	$0.398888\pi$
$168$		0			0
$169$	−7.76393			−0.597226
$170$		0			0
$171$		−	11.4412i		−	0.874933i
$172$		0			0
$173$	−18.1784			−1.38208			−0.691041	−	0.722816i	$-0.742848\pi$
$173$	−18.1784			−1.38208			−0.691041	+	0.722816i	$0.742848\pi$
$174$		0			0
$175$	−6.65841	−	10.5279i	−0.503328	−	0.795832i
$176$		0			0
$177$	−3.70820			−0.278726
$178$		0			0
$179$	−20.6525			−1.54364			−0.771819	−	0.635842i	$-0.780653\pi$
$179$	−20.6525			−1.54364			−0.771819	+	0.635842i	$0.780653\pi$
$180$		0			0
$181$	−0.540182			−0.0401514			−0.0200757	−	0.999798i	$-0.506391\pi$
$181$	−0.540182			−0.0401514			−0.0200757	+	0.999798i	$0.506391\pi$
$182$		0			0
$183$		−	30.1803i		−	2.23099i
$184$		0			0
$185$		−	3.75117i		−	0.275791i
$186$		0			0
$187$			2.16073i			0.158008i
$188$		0			0
$189$	−3.90879	+	2.47214i	−0.284323	+	0.179821i
$190$		0			0
$191$			22.9443i			1.66019i	0.557623	+	0.830095i	$0.311714\pi$
$191$			22.9443i			1.66019i	−0.557623	+	0.830095i	$0.688286\pi$
$192$		0			0
$193$	−11.2361			−0.808790			−0.404395	−	0.914584i	$-0.632518\pi$
$193$	−11.2361			−0.808790			−0.404395	+	0.914584i	$0.632518\pi$
$194$		0			0
$195$		−	2.82843i		−	0.202548i
$196$		0			0
$197$		−	18.9443i		−	1.34972i	−0.737944	−	0.674862i	$-0.764203\pi$
$197$		−	18.9443i		−	1.34972i	0.737944	−	0.674862i	$-0.235797\pi$
$198$		0			0
$199$	6.73722			0.477589			0.238794	−	0.971070i	$-0.423248\pi$
$199$	6.73722			0.477589			0.238794	+	0.971070i	$0.423248\pi$
$200$		0			0
$201$			8.48528i			0.598506i
$202$		0			0
$203$	−4.47214	+	2.82843i	−0.313882	+	0.198517i
$204$		0			0
$205$			5.52786i			0.386083i
$206$		0			0
$207$		−	7.23607i		−	0.502941i
$208$		0			0
$209$		−	6.32456i		−	0.437479i
$210$		0			0
$211$	−1.23607			−0.0850944			−0.0425472	−	0.999094i	$-0.513547\pi$
$211$	−1.23607			−0.0850944			−0.0425472	+	0.999094i	$0.513547\pi$
$212$		0			0
$213$	22.8825			1.56788
$214$		0			0
$215$	6.32456			0.431331
$216$		0			0
$217$	5.52786	+	8.74032i	0.375256	+	0.593332i
$218$		0			0
$219$	32.3607			2.18673
$220$		0			0
$221$		−	4.00000i		−	0.269069i
$222$		0			0
$223$	16.1452			1.08117			0.540583	−	0.841291i	$-0.318204\pi$
$223$	16.1452			1.08117			0.540583	+	0.841291i	$0.318204\pi$
$224$		0			0
$225$	10.5279			0.701858
$226$		0			0
$227$		−	1.87558i		−	0.124487i	−0.998061	−	0.0622434i	$-0.980174\pi$
$227$		−	1.87558i		−	0.124487i	0.998061	−	0.0622434i	$-0.0198255\pi$
$228$		0			0
$229$	−1.20788			−0.0798191			−0.0399096	−	0.999203i	$-0.512707\pi$
$229$	−1.20788			−0.0798191			−0.0399096	+	0.999203i	$0.512707\pi$
$230$		0			0
$231$	6.32456	−	4.00000i	0.416125	−	0.263181i
$232$		0			0
$233$	13.4164			0.878938			0.439469	−	0.898258i	$-0.355167\pi$
$233$	13.4164			0.878938			0.439469	+	0.898258i	$0.355167\pi$
$234$		0			0
$235$	−5.88854			−0.384126
$236$		0			0
$237$	−8.07262			−0.524373
$238$		0			0
$239$			18.6525i			1.20653i	0.797541	+	0.603264i	$0.206134\pi$
$239$			18.6525i			1.20653i	−0.797541	+	0.603264i	$0.793866\pi$
$240$		0			0
$241$		−	1.74806i		−	0.112603i	−0.998414	−	0.0563014i	$-0.982069\pi$
$241$		−	1.74806i		−	0.112603i	0.998414	−	0.0563014i	$-0.0179308\pi$
$242$		0			0
$243$		−	19.2588i		−	1.23545i
$244$		0			0
$245$	1.62054	−	3.41641i	0.103533	−	0.218266i
$246$		0			0
$247$			11.7082i			0.744975i
$248$		0			0
$249$	20.6525			1.30880
$250$		0			0
$251$			19.2588i			1.21561i	0.794088	+	0.607803i	$0.207949\pi$
$251$			19.2588i			1.21561i	−0.794088	+	0.607803i	$0.792051\pi$
$252$		0			0
$253$		−	4.00000i		−	0.251478i
$254$		0			0
$255$	−2.16073			−0.135310
$256$		0			0
$257$			16.9706i			1.05859i	0.848436	+	0.529297i	$0.177544\pi$
$257$			16.9706i			1.05859i	−0.848436	+	0.529297i	$0.822456\pi$
$258$		0			0
$259$	−15.5279	+	9.82068i	−0.964855	+	0.610228i
$260$		0			0
$261$		−	4.47214i		−	0.276818i
$262$		0			0
$263$		−	3.88854i		−	0.239778i	−0.992787	−	0.119889i	$-0.961746\pi$
$263$		−	3.88854i		−	0.239778i	0.992787	−	0.119889i	$-0.0382539\pi$
$264$		0			0
$265$		−	4.57649i		−	0.281132i
$266$		0			0
$267$	−35.4164			−2.16745
$268$		0			0
$269$	19.2588			1.17423			0.587115	−	0.809503i	$-0.300264\pi$
$269$	19.2588			1.17423			0.587115	+	0.809503i	$0.300264\pi$
$270$		0			0
$271$	26.1235			1.58689			0.793446	−	0.608640i	$-0.208285\pi$
$271$	26.1235			1.58689			0.793446	+	0.608640i	$0.208285\pi$
$272$		0			0
$273$	−11.7082	+	7.40492i	−0.708613	+	0.448166i
$274$		0			0
$275$	5.81966			0.350939
$276$		0			0
$277$			18.9443i			1.13825i	0.822251	+	0.569125i	$0.192718\pi$
$277$			18.9443i			1.13825i	−0.822251	+	0.569125i	$0.807282\pi$
$278$		0			0
$279$	−8.74032			−0.523269
$280$		0			0
$281$	3.88854			0.231971			0.115986	−	0.993251i	$-0.462997\pi$
$281$	3.88854			0.231971			0.115986	+	0.993251i	$0.462997\pi$
$282$		0			0
$283$			20.5942i			1.22420i	0.790781	+	0.612099i	$0.209675\pi$
$283$			20.5942i			1.22420i	−0.790781	+	0.612099i	$0.790325\pi$
$284$		0			0
$285$	6.32456			0.374634
$286$		0			0
$287$	22.8825	−	14.4721i	1.35071	−	0.854263i
$288$		0			0
$289$	13.9443			0.820251
$290$		0			0
$291$	−4.00000			−0.234484
$292$		0			0
$293$	28.4118			1.65983			0.829917	−	0.557886i	$-0.188388\pi$
$293$	28.4118			1.65983			0.829917	+	0.557886i	$0.188388\pi$
$294$		0			0
$295$		−	0.875388i		−	0.0509671i
$296$		0			0
$297$		−	2.16073i		−	0.125378i
$298$		0			0
$299$			7.40492i			0.428237i
$300$		0			0
$301$	−16.5579	−	26.1803i	−0.954382	−	1.50901i
$302$		0			0
$303$			26.1803i			1.50402i
$304$		0			0
$305$	7.12461			0.407954
$306$		0			0
$307$			13.6020i			0.776305i	0.921595	+	0.388152i	$0.126887\pi$
$307$			13.6020i			0.776305i	−0.921595	+	0.388152i	$0.873113\pi$
$308$		0			0
$309$		−	31.4164i		−	1.78722i
$310$		0			0
$311$	14.1421			0.801927			0.400963	−	0.916094i	$-0.368675\pi$
$311$	14.1421			0.801927			0.400963	+	0.916094i	$0.368675\pi$
$312$		0			0
$313$		−	8.89794i		−	0.502941i	−0.967865	−	0.251471i	$-0.919086\pi$
$313$		−	8.89794i		−	0.502941i	0.967865	−	0.251471i	$-0.0809142\pi$
$314$		0			0
$315$	1.70820	+	2.70091i	0.0962464	+	0.152179i
$316$		0			0
$317$		−	19.8885i		−	1.11705i	−0.829487	−	0.558526i	$-0.811367\pi$
$317$		−	19.8885i		−	1.11705i	0.829487	−	0.558526i	$-0.188633\pi$
$318$		0			0
$319$		−	2.47214i		−	0.138413i
$320$		0			0
$321$			0.667701i			0.0372674i
$322$		0			0
$323$	8.94427			0.497673
$324$		0			0
$325$	−10.7735			−0.597608
$326$		0			0
$327$	8.07262			0.446417
$328$		0			0
$329$	15.4164	+	24.3755i	0.849934	+	1.34386i
$330$		0			0
$331$	−15.1246			−0.831324			−0.415662	−	0.909519i	$-0.636450\pi$
$331$	−15.1246			−0.831324			−0.415662	+	0.909519i	$0.636450\pi$
$332$		0			0
$333$		−	15.5279i		−	0.850922i
$334$		0			0
$335$	−2.00310			−0.109441
$336$		0			0
$337$	25.1246			1.36862			0.684312	−	0.729189i	$-0.260102\pi$
$337$	25.1246			1.36862			0.684312	+	0.729189i	$0.260102\pi$
$338$		0			0
$339$			5.24419i			0.284825i
$340$		0			0
$341$	−4.83153			−0.261642
$342$		0			0
$343$	−18.3848	+	2.23607i	−0.992685	+	0.120736i
$344$		0			0
$345$	4.00000			0.215353
$346$		0			0
$347$	9.23607			0.495818			0.247909	−	0.968783i	$-0.420257\pi$
$347$	9.23607			0.495818			0.247909	+	0.968783i	$0.420257\pi$
$348$		0			0
$349$	12.9343			0.692355			0.346177	−	0.938169i	$-0.387480\pi$
$349$	12.9343			0.692355			0.346177	+	0.938169i	$0.387480\pi$
$350$		0			0
$351$			4.00000i			0.213504i
$352$		0			0
$353$			16.1452i			0.859324i	0.902990	+	0.429662i	$0.141367\pi$
$353$			16.1452i			0.859324i	−0.902990	+	0.429662i	$0.858633\pi$
$354$		0			0
$355$			5.40182i			0.286699i
$356$		0			0
$357$	5.65685	+	8.94427i	0.299392	+	0.473381i
$358$		0			0
$359$			27.2361i			1.43746i	0.695287	+	0.718732i	$0.255277\pi$
$359$			27.2361i			1.43746i	−0.695287	+	0.718732i	$0.744723\pi$
$360$		0			0
$361$	−7.18034			−0.377913
$362$		0			0
$363$		−	21.6746i		−	1.13762i
$364$		0			0
$365$			7.63932i			0.399860i
$366$		0			0
$367$	5.65685			0.295285			0.147643	−	0.989041i	$-0.452831\pi$
$367$	5.65685			0.295285			0.147643	+	0.989041i	$0.452831\pi$
$368$		0			0
$369$			22.8825i			1.19121i
$370$		0			0
$371$	−18.9443	+	11.9814i	−0.983538	+	0.622044i
$372$		0			0
$373$		−	5.05573i		−	0.261776i	−0.991397	−	0.130888i	$-0.958217\pi$
$373$		−	5.05573i		−	0.261776i	0.991397	−	0.130888i	$-0.0417828\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$		0			0
$377$			4.57649i			0.235701i
$378$		0			0
$379$	−8.29180			−0.425921			−0.212960	−	0.977061i	$-0.568311\pi$
$379$	−8.29180			−0.425921			−0.212960	+	0.977061i	$0.568311\pi$
$380$		0			0
$381$	−1.74806			−0.0895560
$382$		0			0
$383$	−29.2070			−1.49241			−0.746204	−	0.665717i	$-0.768126\pi$
$383$	−29.2070			−1.49241			−0.746204	+	0.665717i	$0.768126\pi$
$384$		0			0
$385$	0.944272	+	1.49302i	0.0481246	+	0.0760916i
$386$		0			0
$387$	26.1803			1.33082
$388$		0			0
$389$		−	28.8328i		−	1.46188i	−0.682441	−	0.730941i	$-0.739081\pi$
$389$		−	28.8328i		−	1.46188i	0.682441	−	0.730941i	$-0.260919\pi$
$390$		0			0
$391$	5.65685			0.286079
$392$		0			0
$393$	−13.2361			−0.667671
$394$		0			0
$395$		−	1.90569i		−	0.0958855i
$396$		0			0
$397$	31.2402			1.56790			0.783951	−	0.620823i	$-0.213201\pi$
$397$	31.2402			1.56790			0.783951	+	0.620823i	$0.213201\pi$
$398$		0			0
$399$	−16.5579	−	26.1803i	−0.828932	−	1.31066i
$400$		0			0
$401$	14.6525			0.731710			0.365855	−	0.930672i	$-0.380777\pi$
$401$	14.6525			0.731710			0.365855	+	0.930672i	$0.380777\pi$
$402$		0			0
$403$	8.94427			0.445546
$404$		0			0
$405$	5.78437			0.287428
$406$		0			0
$407$		−	8.58359i		−	0.425473i
$408$		0			0
$409$		−	5.91189i		−	0.292324i	−0.989261	−	0.146162i	$-0.953308\pi$
$409$		−	5.91189i		−	0.292324i	0.989261	−	0.146162i	$-0.0466921\pi$
$410$		0			0
$411$			28.5393i			1.40774i
$412$		0			0
$413$	−3.62365	+	2.29180i	−0.178308	+	0.112772i
$414$		0			0
$415$			4.87539i			0.239323i
$416$		0			0
$417$	6.18034			0.302653
$418$		0			0
$419$			22.7549i			1.11165i	0.831299	+	0.555826i	$0.187598\pi$
$419$			22.7549i			1.11165i	−0.831299	+	0.555826i	$0.812402\pi$
$420$		0			0
$421$			17.4164i			0.848824i	0.905469	+	0.424412i	$0.139519\pi$
$421$			17.4164i			0.848824i	−0.905469	+	0.424412i	$0.860481\pi$
$422$		0			0
$423$	−24.3755			−1.18518
$424$		0			0
$425$			8.23024i			0.399225i
$426$		0			0
$427$	−18.6525	−	29.4922i	−0.902657	−	1.42723i
$428$		0			0
$429$		−	6.47214i		−	0.312478i
$430$		0			0
$431$		−	2.29180i		−	0.110392i	−0.998476	−	0.0551960i	$-0.982422\pi$
$431$		−	2.29180i		−	0.110392i	0.998476	−	0.0551960i	$-0.0175784\pi$
$432$		0			0
$433$		−	14.3972i		−	0.691884i	−0.938256	−	0.345942i	$-0.887559\pi$
$433$		−	14.3972i		−	0.691884i	0.938256	−	0.345942i	$-0.112441\pi$
$434$		0			0
$435$	2.47214			0.118530
$436$		0			0
$437$	−16.5579			−0.792072
$438$		0			0
$439$	26.7912			1.27868			0.639338	−	0.768926i	$-0.279208\pi$
$439$	26.7912			1.27868			0.639338	+	0.768926i	$0.279208\pi$
$440$		0			0
$441$	6.70820	−	14.1421i	0.319438	−	0.673435i
$442$		0			0
$443$	31.7082			1.50650			0.753251	−	0.657733i	$-0.228485\pi$
$443$	31.7082			1.50650			0.753251	+	0.657733i	$0.228485\pi$
$444$		0			0
$445$		−	8.36068i		−	0.396334i
$446$		0			0
$447$	30.7000			1.45206
$448$		0			0
$449$	30.3607			1.43281			0.716405	−	0.697685i	$-0.245787\pi$
$449$	30.3607			1.43281			0.716405	+	0.697685i	$0.245787\pi$
$450$		0			0
$451$			12.6491i			0.595623i
$452$		0			0
$453$	−25.7109			−1.20800
$454$		0			0
$455$	−1.74806	−	2.76393i	−0.0819505	−	0.129575i
$456$		0			0
$457$	−30.6525			−1.43386			−0.716931	−	0.697144i	$-0.754454\pi$
$457$	−30.6525			−1.43386			−0.716931	+	0.697144i	$0.754454\pi$
$458$		0			0
$459$	3.05573			0.142629
$460$		0			0
$461$	17.7658			0.827435			0.413718	−	0.910405i	$-0.364230\pi$
$461$	17.7658			0.827435			0.413718	+	0.910405i	$0.364230\pi$
$462$		0			0
$463$		−	31.8885i		−	1.48199i	−0.671513	−	0.740993i	$-0.734355\pi$
$463$		−	31.8885i		−	1.48199i	0.671513	−	0.740993i	$-0.265645\pi$
$464$		0			0
$465$		−	4.83153i		−	0.224057i
$466$		0			0
$467$		−	0.540182i		−	0.0249966i	−0.999922	−	0.0124983i	$-0.996022\pi$
$467$		−	0.540182i		−	0.0249966i	0.999922	−	0.0124983i	$-0.00397844\pi$
$468$		0			0
$469$	5.24419	+	8.29180i	0.242154	+	0.382880i
$470$		0			0
$471$			26.1803i			1.20633i
$472$		0			0
$473$	14.4721			0.665430
$474$		0			0
$475$		−	24.0903i		−	1.10534i
$476$		0			0
$477$		−	18.9443i		−	0.867399i
$478$		0			0
$479$	−27.8716			−1.27349			−0.636743	−	0.771076i	$-0.719719\pi$
$479$	−27.8716			−1.27349			−0.636743	+	0.771076i	$0.719719\pi$
$480$		0			0
$481$			15.8902i			0.724531i
$482$		0			0
$483$	−10.4721	−	16.5579i	−0.476499	−	0.753411i
$484$		0			0
$485$		−	0.944272i		−	0.0428772i
$486$		0			0
$487$			34.0689i			1.54381i	0.635739	+	0.771904i	$0.280696\pi$
$487$			34.0689i			1.54381i	−0.635739	+	0.771904i	$0.719304\pi$
$488$		0			0
$489$			39.4404i			1.78355i
$490$		0			0
$491$	−27.7082			−1.25045			−0.625227	−	0.780443i	$-0.714994\pi$
$491$	−27.7082			−1.25045			−0.625227	+	0.780443i	$0.714994\pi$
$492$		0			0
$493$	3.49613			0.157458
$494$		0			0
$495$	−1.49302			−0.0671065
$496$		0			0
$497$	22.3607	−	14.1421i	1.00301	−	0.634361i
$498$		0			0
$499$	25.2361			1.12972			0.564861	−	0.825186i	$-0.308930\pi$
$499$	25.2361			1.12972			0.564861	+	0.825186i	$0.308930\pi$
$500$		0			0
$501$			18.4721i			0.825274i
$502$		0			0
$503$	−28.1266			−1.25411			−0.627053	−	0.778977i	$-0.715739\pi$
$503$	−28.1266			−1.25411			−0.627053	+	0.778977i	$0.715739\pi$
$504$		0			0
$505$	−6.18034			−0.275022
$506$		0			0
$507$		−	17.7658i		−	0.789006i
$508$		0			0
$509$	25.5834			1.13396			0.566981	−	0.823731i	$-0.308111\pi$
$509$	25.5834			1.13396			0.566981	+	0.823731i	$0.308111\pi$
$510$		0			0
$511$	31.6228	−	20.0000i	1.39891	−	0.884748i
$512$		0			0
$513$	−8.94427			−0.394899
$514$		0			0
$515$	7.41641			0.326806
$516$		0			0
$517$	−13.4744			−0.592605
$518$		0			0
$519$		−	41.5967i		−	1.82589i
$520$		0			0
$521$			15.0649i			0.660004i	0.943980	+	0.330002i	$0.107049\pi$
$521$			15.0649i			0.660004i	−0.943980	+	0.330002i	$0.892951\pi$
$522$		0			0
$523$		−	15.3500i		−	0.671209i	−0.942003	−	0.335605i	$-0.891059\pi$
$523$		−	15.3500i		−	0.671209i	0.942003	−	0.335605i	$-0.108941\pi$
$524$		0			0
$525$	24.0903	−	15.2361i	1.05139	−	0.664957i
$526$		0			0
$527$		−	6.83282i		−	0.297642i
$528$		0			0
$529$	12.5279			0.544690
$530$		0			0
$531$		−	3.62365i		−	0.157253i
$532$		0			0
$533$		−	23.4164i		−	1.01428i
$534$		0			0
$535$	−0.157623			−0.00681463
$536$		0			0
$537$		−	47.2579i		−	2.03933i
$538$		0			0
$539$	3.70820	−	7.81758i	0.159724	−	0.336727i
$540$		0			0
$541$			8.47214i			0.364246i	0.983276	+	0.182123i	$0.0582968\pi$
$541$			8.47214i			0.364246i	−0.983276	+	0.182123i	$0.941703\pi$
$542$		0			0
$543$		−	1.23607i		−	0.0530448i
$544$		0			0
$545$			1.90569i			0.0816307i
$546$		0			0
$547$	3.12461			0.133599			0.0667994	−	0.997766i	$-0.478721\pi$
$547$	3.12461			0.133599			0.0667994	+	0.997766i	$0.478721\pi$
$548$		0			0
$549$	29.4922			1.25869
$550$		0			0
$551$	−10.2333			−0.435955
$552$		0			0
$553$	−7.88854	+	4.98915i	−0.335455	+	0.212160i
$554$		0			0
$555$	8.58359			0.364353
$556$		0			0
$557$		−	42.3607i		−	1.79488i	−0.441137	−	0.897440i	$-0.645425\pi$
$557$		−	42.3607i		−	1.79488i	0.441137	−	0.897440i	$-0.354575\pi$
$558$		0			0
$559$	−26.7912			−1.13315
$560$		0			0
$561$	−4.94427			−0.208747
$562$		0			0
$563$		−	35.8167i		−	1.50949i	−0.656016	−	0.754747i	$-0.727760\pi$
$563$		−	35.8167i		−	1.50949i	0.656016	−	0.754747i	$-0.272240\pi$
$564$		0			0
$565$	−1.23799			−0.0520825
$566$		0			0
$567$	−15.1437	−	23.9443i	−0.635975	−	1.00556i
$568$		0			0
$569$	34.0689			1.42824			0.714121	−	0.700022i	$-0.246827\pi$
$569$	34.0689			1.42824			0.714121	+	0.700022i	$0.246827\pi$
$570$		0			0
$571$	14.1803			0.593429			0.296714	−	0.954966i	$-0.404109\pi$
$571$	14.1803			0.593429			0.296714	+	0.954966i	$0.404109\pi$
$572$		0			0
$573$	−52.5021			−2.19331
$574$		0			0
$575$		−	15.2361i		−	0.635388i
$576$		0			0
$577$			35.2765i			1.46858i	0.678835	+	0.734291i	$0.262485\pi$
$577$			35.2765i			1.46858i	−0.678835	+	0.734291i	$0.737515\pi$
$578$		0			0
$579$		−	25.7109i		−	1.06851i
$580$		0			0
$581$	20.1815	−	12.7639i	0.837272	−	0.529537i
$582$		0			0
$583$		−	10.4721i		−	0.433712i
$584$		0			0
$585$	2.76393			0.114275
$586$		0			0
$587$		−	8.86784i		−	0.366015i	−0.983112	−	0.183007i	$-0.941417\pi$
$587$		−	8.86784i		−	0.366015i	0.983112	−	0.183007i	$-0.0585833\pi$
$588$		0			0
$589$			20.0000i			0.824086i
$590$		0			0
$591$	43.3491			1.78315
$592$		0			0
$593$		−	11.3137i		−	0.464598i	−0.972644	−	0.232299i	$-0.925375\pi$
$593$		−	11.3137i		−	0.464598i	0.972644	−	0.232299i	$-0.0746248\pi$
$594$		0			0
$595$	−2.11146	+	1.33540i	−0.0865613	+	0.0547462i
$596$		0			0
$597$			15.4164i			0.630952i
$598$		0			0
$599$		−	3.52786i		−	0.144145i	−0.997399	−	0.0720723i	$-0.977039\pi$
$599$		−	3.52786i		−	0.144145i	0.997399	−	0.0720723i	$-0.0229612\pi$
$600$		0			0
$601$		−	24.6305i		−	1.00470i	−0.864664	−	0.502350i	$-0.832469\pi$
$601$		−	24.6305i		−	1.00470i	0.864664	−	0.502350i	$-0.167531\pi$
$602$		0			0
$603$	−8.29180			−0.337668
$604$		0			0
$605$	5.11667			0.208022
$606$		0			0
$607$	−23.1375			−0.939122			−0.469561	−	0.882900i	$-0.655588\pi$
$607$	−23.1375			−0.939122			−0.469561	+	0.882900i	$0.655588\pi$
$608$		0			0
$609$	−6.47214	−	10.2333i	−0.262264	−	0.414676i
$610$		0			0
$611$	24.9443			1.00914
$612$		0			0
$613$			9.41641i			0.380325i	0.981753	+	0.190163i	$0.0609015\pi$
$613$			9.41641i			0.380325i	−0.981753	+	0.190163i	$0.939098\pi$
$614$		0			0
$615$	−12.6491			−0.510061
$616$		0			0
$617$	−7.23607			−0.291313			−0.145657	−	0.989335i	$-0.546529\pi$
$617$	−7.23607			−0.291313			−0.145657	+	0.989335i	$0.546529\pi$
$618$		0			0
$619$		−	4.70401i		−	0.189070i	−0.995522	−	0.0945351i	$-0.969864\pi$
$619$		−	4.70401i		−	0.189070i	0.995522	−	0.0945351i	$-0.0301365\pi$
$620$		0			0
$621$	−5.65685			−0.227002
$622$		0			0
$623$	−34.6088	+	21.8885i	−1.38657	+	0.876946i
$624$		0			0
$625$	20.7082			0.828328
$626$		0			0
$627$	14.4721			0.577961
$628$		0			0
$629$	12.1390			0.484015
$630$		0			0
$631$		−	0.472136i		−	0.0187954i	−0.999956	−	0.00939772i	$-0.997009\pi$
$631$		−	0.472136i		−	0.0187954i	0.999956	−	0.00939772i	$-0.00299143\pi$
$632$		0			0
$633$		−	2.82843i		−	0.112420i
$634$		0			0
$635$		−	0.412662i		−	0.0163760i
$636$		0			0
$637$	−6.86474	+	14.4721i	−0.271991	+	0.573407i
$638$		0			0
$639$			22.3607i			0.884575i
$640$		0			0
$641$	−16.1803			−0.639085			−0.319543	−	0.947572i	$-0.603529\pi$
$641$	−16.1803			−0.639085			−0.319543	+	0.947572i	$0.603529\pi$
$642$		0			0
$643$		−	16.6854i		−	0.658009i	−0.944328	−	0.329004i	$-0.893287\pi$
$643$		−	16.6854i		−	0.658009i	0.944328	−	0.329004i	$-0.106713\pi$
$644$		0			0
$645$			14.4721i			0.569840i
$646$		0			0
$647$	2.41577			0.0949735			0.0474868	−	0.998872i	$-0.484879\pi$
$647$	2.41577			0.0949735			0.0474868	+	0.998872i	$0.484879\pi$
$648$		0			0
$649$		−	2.00310i		−	0.0786287i
$650$		0			0
$651$	−20.0000	+	12.6491i	−0.783862	+	0.495758i
$652$		0			0
$653$			13.4164i			0.525025i	0.964929	+	0.262512i	$0.0845510\pi$
$653$			13.4164i			0.525025i	−0.964929	+	0.262512i	$0.915449\pi$
$654$		0			0
$655$		−	3.12461i		−	0.122089i
$656$		0			0
$657$			31.6228i			1.23372i
$658$		0			0
$659$	−41.0132			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$659$	−41.0132			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$660$		0			0
$661$	−26.5061			−1.03097			−0.515484	−	0.856899i	$-0.672388\pi$
$661$	−26.5061			−1.03097			−0.515484	+	0.856899i	$0.672388\pi$
$662$		0			0
$663$	9.15298			0.355472
$664$		0			0
$665$	6.18034	−	3.90879i	0.239663	−	0.151576i
$666$		0			0
$667$	−6.47214			−0.250602
$668$		0			0
$669$			36.9443i			1.42835i
$670$		0			0
$671$	16.3029			0.629365
$672$		0			0
$673$	−40.8328			−1.57399			−0.786995	−	0.616960i	$-0.788364\pi$
$673$	−40.8328			−1.57399			−0.786995	+	0.616960i	$0.788364\pi$
$674$		0			0
$675$		−	8.23024i		−	0.316782i
$676$		0			0
$677$	32.5756			1.25198			0.625991	−	0.779830i	$-0.284694\pi$
$677$	32.5756			1.25198			0.625991	+	0.779830i	$0.284694\pi$
$678$		0			0
$679$	−3.90879	+	2.47214i	−0.150006	+	0.0948719i
$680$		0			0
$681$	4.29180			0.164462
$682$		0			0
$683$	10.1803			0.389540			0.194770	−	0.980849i	$-0.437604\pi$
$683$	10.1803			0.389540			0.194770	+	0.980849i	$0.437604\pi$
$684$		0			0
$685$	−6.73722			−0.257416
$686$		0			0
$687$		−	2.76393i		−	0.105451i
$688$		0			0
$689$			19.3863i			0.738560i
$690$		0			0
$691$			6.60970i			0.251445i	0.992065	+	0.125722i	$0.0401249\pi$
$691$			6.60970i			0.251445i	−0.992065	+	0.125722i	$0.959875\pi$
$692$		0			0
$693$	3.90879	+	6.18034i	0.148483	+	0.234772i
$694$		0			0
$695$			1.45898i			0.0553423i
$696$		0			0
$697$	−17.8885			−0.677577
$698$		0			0
$699$			30.7000i			1.16118i
$700$		0			0
$701$		−	31.5279i		−	1.19079i	−0.803433	−	0.595395i	$-0.796995\pi$
$701$		−	31.5279i		−	1.19079i	0.803433	−	0.595395i	$-0.203005\pi$
$702$		0			0
$703$	−35.5316			−1.34010
$704$		0			0
$705$		−	13.4744i		−	0.507477i
$706$		0			0
$707$	16.1803	+	25.5834i	0.608524	+	0.962161i
$708$		0			0
$709$		−	12.4721i		−	0.468401i	−0.972188	−	0.234200i	$-0.924753\pi$
$709$		−	12.4721i		−	0.468401i	0.972188	−	0.234200i	$-0.0752472\pi$
$710$		0			0
$711$		−	7.88854i		−	0.295844i
$712$		0			0
$713$			12.6491i			0.473713i
$714$		0			0
$715$	1.52786			0.0571389
$716$		0			0
$717$	−42.6814			−1.59397
$718$		0			0
$719$	45.3523			1.69135			0.845677	−	0.533695i	$-0.179197\pi$
$719$	45.3523			1.69135			0.845677	+	0.533695i	$0.179197\pi$
$720$		0			0
$721$	−19.4164	−	30.7000i	−0.723105	−	1.14333i
$722$		0			0
$723$	4.00000			0.148762
$724$		0			0
$725$		−	9.41641i		−	0.349717i
$726$		0			0
$727$	2.41577			0.0895958			0.0447979	−	0.998996i	$-0.485736\pi$
$727$	2.41577			0.0895958			0.0447979	+	0.998996i	$0.485736\pi$
$728$		0			0
$729$	11.9443			0.442380
$730$		0			0
$731$			20.4667i			0.756988i
$732$		0			0
$733$	−7.02236			−0.259377			−0.129688	−	0.991555i	$-0.541398\pi$
$733$	−7.02236			−0.259377			−0.129688	+	0.991555i	$0.541398\pi$
$734$		0			0
$735$	7.81758	+	3.70820i	0.288356	+	0.136779i
$736$		0			0
$737$	−4.58359			−0.168839
$738$		0			0
$739$	2.40325			0.0884051			0.0442025	−	0.999023i	$-0.485925\pi$
$739$	2.40325			0.0884051			0.0442025	+	0.999023i	$0.485925\pi$
$740$		0			0
$741$	−26.7912			−0.984201
$742$		0			0
$743$		−	17.7082i		−	0.649651i	−0.945774	−	0.324825i	$-0.894694\pi$
$743$		−	17.7082i		−	0.649651i	0.945774	−	0.324825i	$-0.105306\pi$
$744$		0			0
$745$			7.24730i			0.265520i
$746$		0			0
$747$			20.1815i			0.738404i
$748$		0			0
$749$	0.412662	+	0.652476i	0.0150783	+	0.0238409i
$750$		0			0
$751$			52.5410i			1.91725i	0.284674	+	0.958625i	$0.408115\pi$
$751$			52.5410i			1.91725i	−0.284674	+	0.958625i	$0.591885\pi$
$752$		0			0
$753$	−44.0689			−1.60596
$754$		0			0
$755$		−	6.06952i		−	0.220892i
$756$		0			0
$757$			22.3607i			0.812713i	0.913715	+	0.406356i	$0.133201\pi$
$757$			22.3607i			0.812713i	−0.913715	+	0.406356i	$0.866799\pi$
$758$		0			0
$759$	9.15298			0.332232
$760$		0			0
$761$			8.07262i			0.292632i	0.989238	+	0.146316i	$0.0467417\pi$
$761$			8.07262i			0.292632i	−0.989238	+	0.146316i	$0.953258\pi$
$762$		0			0
$763$	7.88854	−	4.98915i	0.285584	−	0.180619i
$764$		0			0
$765$		−	2.11146i		−	0.0763399i
$766$		0			0
$767$			3.70820i			0.133895i
$768$		0			0
$769$		−	8.23024i		−	0.296790i	−0.988928	−	0.148395i	$-0.952589\pi$
$769$		−	8.23024i		−	0.296790i	0.988928	−	0.148395i	$-0.0474107\pi$
$770$		0			0
$771$	−38.8328			−1.39853
$772$		0			0
$773$	−49.8012			−1.79123			−0.895613	−	0.444835i	$-0.853262\pi$
$773$	−49.8012			−1.79123			−0.895613	+	0.444835i	$0.853262\pi$
$774$		0			0
$775$	−18.4034			−0.661069
$776$		0			0
$777$	−22.4721	−	35.5316i	−0.806183	−	1.27469i
$778$		0			0
$779$	52.3607			1.87602
$780$		0			0
$781$			12.3607i			0.442300i
$782$		0			0
$783$	−3.49613			−0.124941
$784$		0			0
$785$	−6.18034			−0.220586
$786$		0			0
$787$			10.7735i			0.384035i	0.981392	+	0.192017i	$0.0615030\pi$
$787$			10.7735i			0.384035i	−0.981392	+	0.192017i	$0.938497\pi$
$788$		0			0
$789$	8.89794			0.316775
$790$		0			0
$791$	3.24109	+	5.12461i	0.115240	+	0.182210i
$792$		0			0
$793$	−30.1803			−1.07174
$794$		0			0
$795$	10.4721			0.371408
$796$		0			0
$797$	13.6020			0.481806			0.240903	−	0.970549i	$-0.422556\pi$
$797$	13.6020			0.481806			0.240903	+	0.970549i	$0.422556\pi$
$798$		0			0
$799$		−	19.0557i		−	0.674143i
$800$		0			0
$801$		−	34.6088i		−	1.22284i
$802$		0			0
$803$			17.4806i			0.616878i
$804$		0			0
$805$	3.90879	−	2.47214i	0.137767	−	0.0871313i
$806$		0			0
$807$			44.0689i			1.55130i
$808$		0			0
$809$	2.06888			0.0727381			0.0363690	−	0.999338i	$-0.488421\pi$
$809$	2.06888			0.0727381			0.0363690	+	0.999338i	$0.488421\pi$
$810$		0			0
$811$			36.7394i			1.29010i	0.764142	+	0.645048i	$0.223163\pi$
$811$			36.7394i			1.29010i	−0.764142	+	0.645048i	$0.776837\pi$
$812$		0			0
$813$			59.7771i			2.09647i
$814$		0			0
$815$	−9.31061			−0.326136
$816$		0			0
$817$		−	59.9070i		−	2.09588i
$818$		0			0
$819$	−7.23607	−	11.4412i	−0.252849	−	0.399789i
$820$		0			0
$821$		−	11.8885i		−	0.414913i	−0.978244	−	0.207457i	$-0.933481\pi$
$821$		−	11.8885i		−	0.414913i	0.978244	−	0.207457i	$-0.0665186\pi$
$822$		0			0
$823$		−	16.8328i		−	0.586755i	−0.955997	−	0.293378i	$-0.905221\pi$
$823$		−	16.8328i		−	0.586755i	0.955997	−	0.293378i	$-0.0947793\pi$
$824$		0			0
$825$			13.3168i			0.463632i
$826$		0			0
$827$	−24.0689			−0.836957			−0.418479	−	0.908227i	$-0.637436\pi$
$827$	−24.0689			−0.836957			−0.418479	+	0.908227i	$0.637436\pi$
$828$		0			0
$829$	−35.8167			−1.24397			−0.621983	−	0.783031i	$-0.713673\pi$
$829$	−35.8167			−1.24397			−0.621983	+	0.783031i	$0.713673\pi$
$830$		0			0
$831$	−43.3491			−1.50377
$832$		0			0
$833$	11.0557	+	5.24419i	0.383058	+	0.181700i
$834$		0			0
$835$	−4.36068			−0.150908
$836$		0			0
$837$			6.83282i			0.236177i
$838$		0			0
$839$	54.6629			1.88717			0.943586	−	0.331128i	$-0.107429\pi$
$839$	54.6629			1.88717			0.943586	+	0.331128i	$0.107429\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$		0			0
$843$			8.89794i			0.306461i
$844$		0			0
$845$	4.19393			0.144276
$846$		0			0
$847$	−13.3956	−	21.1803i	−0.460279	−	0.727765i
$848$		0			0
$849$	−47.1246			−1.61731
$850$		0			0
$851$	−22.4721			−0.770335
$852$		0			0
$853$	−4.19393			−0.143598			−0.0717988	−	0.997419i	$-0.522874\pi$
$853$	−4.19393			−0.143598			−0.0717988	+	0.997419i	$0.522874\pi$
$854$		0			0
$855$			6.18034i			0.211363i
$856$		0			0
$857$			43.3491i			1.48078i	0.672178	+	0.740389i	$0.265359\pi$
$857$			43.3491i			1.48078i	−0.672178	+	0.740389i	$0.734641\pi$
$858$		0			0
$859$		−	18.6885i		−	0.637644i	−0.947815	−	0.318822i	$-0.896713\pi$
$859$		−	18.6885i		−	0.637644i	0.947815	−	0.318822i	$-0.103287\pi$
$860$		0			0
$861$	33.1158	+	52.3607i	1.12858	+	1.78445i
$862$		0			0
$863$			10.0000i			0.340404i	0.985409	+	0.170202i	$0.0544420\pi$
$863$			10.0000i			0.340404i	−0.985409	+	0.170202i	$0.945558\pi$
$864$		0			0
$865$	9.81966			0.333878
$866$		0			0
$867$			31.9079i			1.08365i
$868$		0			0
$869$		−	4.36068i		−	0.147926i
$870$		0			0
$871$	8.48528			0.287513
$872$		0			0
$873$		−	3.90879i		−	0.132293i
$874$		0			0
$875$	7.41641	+	11.7264i	0.250720	+	0.396424i
$876$		0			0
$877$			30.0000i			1.01303i	0.862232	+	0.506514i	$0.169066\pi$
$877$			30.0000i			1.01303i	−0.862232	+	0.506514i	$0.830934\pi$
$878$		0			0
$879$			65.0132i			2.19284i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−32.2918			−1.08671			−0.543353	−	0.839505i	$-0.682845\pi$
$883$	−32.2918			−1.08671			−0.543353	+	0.839505i	$0.682845\pi$
$884$		0			0
$885$	2.00310			0.0673336
$886$		0			0
$887$	2.41577			0.0811135			0.0405567	−	0.999177i	$-0.487087\pi$
$887$	2.41577			0.0811135			0.0405567	+	0.999177i	$0.487087\pi$
$888$		0			0
$889$	−1.70820	+	1.08036i	−0.0572913	+	0.0362342i
$890$		0			0
$891$	13.2361			0.443425
$892$		0			0
$893$			55.7771i			1.86651i
$894$		0			0
$895$	11.1561			0.372907
$896$		0			0
$897$	−16.9443			−0.565753
$898$		0			0
$899$			7.81758i			0.260731i
$900$		0			0
$901$	14.8098			0.493387
$902$		0			0
$903$	59.9070	−	37.8885i	1.99358	−	1.26085i
$904$		0			0
$905$	0.291796			0.00969963
$906$		0			0
$907$	−0.291796			−0.00968893			−0.00484446	−	0.999988i	$-0.501542\pi$
$907$	−0.291796			−0.00968893			−0.00484446	+	0.999988i	$0.501542\pi$
$908$		0			0
$909$	−25.5834			−0.848547
$910$		0			0
$911$		−	25.1246i		−	0.832416i	−0.909270	−	0.416208i	$-0.863359\pi$
$911$		−	25.1246i		−	0.832416i	0.909270	−	0.416208i	$-0.136641\pi$
$912$		0			0
$913$			11.1561i			0.369213i
$914$		0			0
$915$			16.3029i			0.538956i
$916$		0			0
$917$	−12.9343	+	8.18034i	−0.427127	+	0.270139i
$918$		0			0
$919$		−	6.94427i		−	0.229070i	−0.993419	−	0.114535i	$-0.963462\pi$
$919$		−	6.94427i		−	0.229070i	0.993419	−	0.114535i	$-0.0365379\pi$
$920$		0			0
$921$	−31.1246			−1.02559
$922$		0			0
$923$		−	22.8825i		−	0.753185i
$924$		0			0
$925$		−	32.6950i		−	1.07501i
$926$		0			0
$927$	30.7000			1.00832
$928$		0			0
$929$		−	45.3523i		−	1.48796i	−0.668202	−	0.743980i	$-0.732936\pi$
$929$		−	45.3523i		−	1.48796i	0.668202	−	0.743980i	$-0.267064\pi$
$930$		0			0
$931$	−32.3607	−	15.3500i	−1.06058	−	0.503077i
$932$		0			0
$933$			32.3607i			1.05944i
$934$		0			0
$935$		−	1.16718i		−	0.0381710i
$936$		0			0
$937$		−	45.0972i		−	1.47326i	−0.676295	−	0.736631i	$-0.736416\pi$
$937$		−	45.0972i		−	1.47326i	0.676295	−	0.736631i	$-0.263584\pi$
$938$		0			0
$939$	20.3607			0.664446
$940$		0			0
$941$	−39.9805			−1.30333			−0.651664	−	0.758508i	$-0.725929\pi$
$941$	−39.9805			−1.30333			−0.651664	+	0.758508i	$0.725929\pi$
$942$		0			0
$943$	33.1158			1.07840
$944$		0			0
$945$	2.11146	−	1.33540i	0.0686857	−	0.0434406i
$946$		0			0
$947$	−15.1246			−0.491484			−0.245742	−	0.969335i	$-0.579032\pi$
$947$	−15.1246			−0.491484			−0.245742	+	0.969335i	$0.579032\pi$
$948$		0			0
$949$		−	32.3607i		−	1.05047i
$950$		0			0
$951$	45.5099			1.47576
$952$		0			0
$953$	1.05573			0.0341984			0.0170992	−	0.999854i	$-0.494557\pi$
$953$	1.05573			0.0341984			0.0170992	+	0.999854i	$0.494557\pi$
$954$		0			0
$955$		−	12.3941i		−	0.401063i
$956$		0			0
$957$	5.65685			0.182860
$958$		0			0
$959$	17.6383	+	27.8885i	0.569569	+	0.900568i
$960$		0			0
$961$	−15.7214			−0.507141
$962$		0			0
$963$	−0.652476			−0.0210257
$964$		0			0
$965$	6.06952			0.195385
$966$		0			0
$967$			17.3475i			0.557859i	0.960312	+	0.278929i	$0.0899795\pi$
$967$			17.3475i			0.557859i	−0.960312	+	0.278929i	$0.910020\pi$
$968$		0			0
$969$			20.4667i			0.657485i
$970$		0			0
$971$		−	20.1815i		−	0.647657i	−0.946116	−	0.323828i	$-0.895030\pi$
$971$		−	20.1815i		−	0.647657i	0.946116	−	0.323828i	$-0.104970\pi$
$972$		0			0
$973$	6.03941	−	3.81966i	0.193615	−	0.122453i
$974$		0			0
$975$		−	24.6525i		−	0.789511i
$976$		0			0
$977$	33.4164			1.06909			0.534543	−	0.845141i	$-0.320484\pi$
$977$	33.4164			1.06909			0.534543	+	0.845141i	$0.320484\pi$
$978$		0			0
$979$		−	19.1313i		−	0.611439i
$980$		0			0
$981$			7.88854i			0.251862i
$982$		0			0
$983$	1.90569			0.0607820			0.0303910	−	0.999538i	$-0.490325\pi$
$983$	1.90569			0.0607820			0.0303910	+	0.999538i	$0.490325\pi$
$984$		0			0
$985$			10.2333i			0.326061i
$986$		0			0
$987$	−55.7771	+	35.2765i	−1.77540	+	1.12286i
$988$		0			0
$989$		−	37.8885i		−	1.20479i
$990$		0			0
$991$			34.3607i			1.09150i	0.837947	+	0.545751i	$0.183756\pi$
$991$			34.3607i			1.09150i	−0.837947	+	0.545751i	$0.816244\pi$
$992$		0			0
$993$		−	34.6088i		−	1.09828i
$994$		0			0
$995$	−3.63932			−0.115374
$996$		0			0
$997$	−37.9774			−1.20276			−0.601379	−	0.798964i	$-0.705382\pi$
$997$	−37.9774			−1.20276			−0.601379	+	0.798964i	$0.705382\pi$
$998$		0			0
$999$	−12.1390			−0.384062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.e.i.895.7		8
4.3	odd	2		1792.2.e.h.895.1		8
7.6	odd	2	inner	1792.2.e.i.895.2		8
8.3	odd	2	inner	1792.2.e.i.895.8		8
8.5	even	2		1792.2.e.h.895.2		8
16.3	odd	4		896.2.f.b.895.8	yes	8
16.5	even	4		896.2.f.a.895.7	yes	8
16.11	odd	4		896.2.f.a.895.1	✓	8
16.13	even	4		896.2.f.b.895.2	yes	8
28.27	even	2		1792.2.e.h.895.8		8
56.13	odd	2		1792.2.e.h.895.7		8
56.27	even	2	inner	1792.2.e.i.895.1		8
112.13	odd	4		896.2.f.b.895.7	yes	8
112.27	even	4		896.2.f.a.895.8	yes	8
112.69	odd	4		896.2.f.a.895.2	yes	8
112.83	even	4		896.2.f.b.895.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.f.a.895.1	✓	8	16.11	odd	4
896.2.f.a.895.2	yes	8	112.69	odd	4
896.2.f.a.895.7	yes	8	16.5	even	4
896.2.f.a.895.8	yes	8	112.27	even	4
896.2.f.b.895.1	yes	8	112.83	even	4
896.2.f.b.895.2	yes	8	16.13	even	4
896.2.f.b.895.7	yes	8	112.13	odd	4
896.2.f.b.895.8	yes	8	16.3	odd	4
1792.2.e.h.895.1		8	4.3	odd	2
1792.2.e.h.895.2		8	8.5	even	2
1792.2.e.h.895.7		8	56.13	odd	2
1792.2.e.h.895.8		8	28.27	even	2
1792.2.e.i.895.1		8	56.27	even	2	inner
1792.2.e.i.895.2		8	7.6	odd	2	inner
1792.2.e.i.895.7		8	1.1	even	1	trivial
1792.2.e.i.895.8		8	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.28825i q^{3} -0.540182 q^{5} +(1.41421 + 2.23607i) q^{7} -2.23607 q^{9} +O(q^{10})\)	\(q+2.28825i q^{3} -0.540182 q^{5} +(1.41421 + 2.23607i) q^{7} -2.23607 q^{9} -1.23607 q^{11} +2.28825 q^{13} -1.23607i q^{15} -1.74806i q^{17} +5.11667i q^{19} +(-5.11667 + 3.23607i) q^{21} +3.23607i q^{23} -4.70820 q^{25} +1.74806i q^{27} +2.00000i q^{29} +3.90879 q^{31} -2.82843i q^{33} +(-0.763932 - 1.20788i) q^{35} +6.94427i q^{37} +5.23607i q^{39} -10.2333i q^{41} -11.7082 q^{43} +1.20788 q^{45} +10.9010 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.00000 q^{51} +8.47214i q^{53} +0.667701 q^{55} -11.7082 q^{57} +1.62054i q^{59} -13.1893 q^{61} +(-3.16228 - 5.00000i) q^{63} -1.23607 q^{65} +3.70820 q^{67} -7.40492 q^{69} -10.0000i q^{71} -14.1421i q^{73} -10.7735i q^{75} +(-1.74806 - 2.76393i) q^{77} +3.52786i q^{79} -10.7082 q^{81} -9.02546i q^{83} +0.944272i q^{85} -4.57649 q^{87} +15.4775i q^{89} +(3.23607 + 5.11667i) q^{91} +8.94427i q^{93} -2.76393i q^{95} +1.74806i q^{97} +2.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} + 16 q^{25} - 24 q^{35} - 40 q^{43} - 24 q^{49} + 32 q^{51} - 40 q^{57} + 8 q^{65} - 24 q^{67} - 32 q^{81} + 8 q^{91} + 40 q^{99}+O(q^{100}) \) 8 * q + 8 * q^11 + 16 * q^25 - 24 * q^35 - 40 * q^43 - 24 * q^49 + 32 * q^51 - 40 * q^57 + 8 * q^65 - 24 * q^67 - 32 * q^81 + 8 * q^91 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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